# How to find the least common denominator

#### Smrithi

Help me to find the lcd of 2/3, 5/12, 4/10

#### ChipB

MHF Helper
Take a look at the denominators, and convert each into its prime factors. The LCD is the smallest number made up of those prime factors that includes all the prime factors of each. So to get started - what are the prime factors of each denominator? Next, how can you combine them to make one number that is divisible by all three seuts of prime factors?

#### HallsofIvy

MHF Helper
3; 12= 4*3= 2*2*3; 10= 2*5.

So any common denominator will have to have two "2"s because of the "4" in "12", a "3", and "5".

#### Archie

For any pair of positive integers, $a$ and $b$, you can use Euclid's Algorithm to determine the greatest common divisor, $\gcd{(a, b)}$. The least common multiple is then given by$$ab \over \gcd{(a, b)}$$

With three numbers, $a$, $b$ and $c$ we find the least common multiple $m$ of two of them ($a$ and $b$, say) , and then find the least common multiple of $m$ and $c$.

Euclid's Algorithm is very simple. Example: 208 and 117
\DeclareMathOperator{\lcm}{lcm} \begin{aligned} 208 &= 1 \times 117 + 91 \\ 117 &= 1 \times 91 + 26 \\ 91 &= 3 \times 26 + 13 \\ 26 &= 2 \times 13 \\ \gcd{(117,208)} &= 13 \\ \lcm{(117,208)} &= {117 \times 208 \over \gcd{(117,208)}} = {24336 \over 13} = 1872 \end{aligned}

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#### johnsomeone

In addition to the other comments, you should reduce all your fractions first. That 4/10 should be set equal to 2/5 before you begin. In this case, it won't matter, but in some it will.

If that 5/12 wasn't included (so looking for lcd of 2/3 and 4/10), and you had neglected to reduce 4/10 to 2/5 before proceeding, then you'd calculate that the lcd = 30, but in fact it's 15.
The computed 30 is A common denominator for 2/3 and 4/10, but it's not the LEAST common denominator.

Yes Archie, I initially wrote 4/10 = 1/5. I've got a handle on this math stuff, except for the grade school material. : )
Thanks for catching my mistake.

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